High performance computing for spherical conformal and Riemann mappings

A classical way of finding the harmonic map is to minimize the harmonic energy by the time evolution of the solution of a nonlinear heat diffusion equation. To arrive at the desired harmonic map, which is a steady-state of this equation, we propose an efficient quasi-implicit Euler method and analyze its convergence under some simplifications. If the initial map is not close to the steady-state solution, it is difficult to find the stability region of the time steps. To remedy this drawback, we propose a two-phase approach for the quasi-implicit Euler method. In order to accelerate the convergence, a variant time step scheme and a heuristic method to determine an initial time step are developed. Numerical results clearly demonstrate that the proposed method achieves high performance for computing the spherical conformal and Riemann mappings.

[1]  T. Chan,et al.  Genus zero surface conformal mapping and its application to brain surface mapping. , 2004, IEEE transactions on medical imaging.

[2]  Bruno Lévy,et al.  Least squares conformal maps for automatic texture atlas generation , 2002, ACM Trans. Graph..

[3]  Pedro V. Sander,et al.  Texture mapping progressive meshes , 2001, SIGGRAPH.

[4]  Philip L. Bowers,et al.  Coordinate systems for conformal cerebellar flat maps , 2000, NeuroImage.

[5]  Klaus Ritter,et al.  Optimal approximation of stochastic differential equations by adaptive step-size control , 2000, Math. Comput..

[6]  R. Spigler,et al.  Convergence analysis of the semi-implicit euler method for abstract evolution equations , 1995 .

[7]  Guillermo Sapiro,et al.  Conformal Surface Parameterization for Texture Mapping , 1999 .

[8]  Mingzhu Liu,et al.  CONVERGENCE AND STABILITY OF THE SEMI-IMPLICIT EULER METHOD WITH VARIABLE STEPSIZE FOR A LINEAR STOCHASTIC PANTOGRAPH DIFFERENTIAL EQUATION , 2010 .

[9]  C. Kelley,et al.  Convergence Analysis of Pseudo-Transient Continuation , 1998 .

[10]  Pablo Groisman,et al.  Totally Discrete Explicit and Semi-implicit Euler Methods for a Blow-up Problem in Several Space Dimensions , 2005, Computing.

[11]  Tony DeRose,et al.  Multiresolution analysis of arbitrary meshes , 1995, SIGGRAPH.

[12]  M. Floater.  Mean value coordinates , 2003, Computer Aided Geometric Design.

[13]  Shing-Tung Yau,et al.  Computational Conformal Geometry , 2016 .

[14]  Mark Meyer,et al.  Intrinsic Parameterizations of Surface Meshes , 2002, Comput. Graph. Forum.

[15]  Alla Sheffer,et al.  Parameterization of Faceted Surfaces for Meshing using Angle-Based Flattening , 2001, Engineering with Computers.

[16]  Bruno Lévy,et al.  ABF++: fast and robust angle based flattening , 2005, TOGS.

[17]  Hiromasa Suzuki,et al.  Three-dimensional geometric metamorphosis based on harmonic maps , 1998, The Visual Computer.

[18]  Xin Li,et al.  Curve Space: Classifying Curves On Surfaces , 2007, Commun. Inf. Syst..

[19]  Yalin Wang,et al.  Computing Conformal Invariants: Period Matrices , 2003, Commun. Inf. Syst..

[20]  Ron Kikinis,et al.  Conformal Geometry and Brain Flattening , 1999, MICCAI.

[21]  Ming-Yong Pang,et al.  Survey on Planar Parameterization of Triangular Meshes , 2010, 2010 International Conference on Measuring Technology and Mechatronics Automation.

[22]  Z. Wen,et al.  FAST AND ROBUST ALGORITHMS FOR HARMONIC ENERGY MINIMIZATION ON GENUS-0 SURFACES , 2011 .

[23]  Mark Meyer,et al.  Interactive geometry remeshing , 2002, SIGGRAPH.

[24]  Shing-Tung Yau,et al.  Surface classification using conformal structures , 2003, Proceedings Ninth IEEE International Conference on Computer Vision.

[25]  Martin Isenburg,et al.  Connectivity shapes , 2001, Proceedings Visualization, 2001. VIS '01..

[26]  Ulrich Pinkall,et al.  Computing Discrete Minimal Surfaces and Their Conjugates , 1993, Exp. Math..

[27]  Lars Diening,et al.  Semi-implicit Euler Scheme for Generalized Newtonian Fluids , 2006, SIAM J. Numer. Anal..

[28]  S. Yau,et al.  Global conformal surface parameterization , 2003 .

[29]  C. T. Kelley,et al.  Pseudo-Transient Continuation for Nonsmooth Nonlinear Equations , 2005, SIAM J. Numer. Anal..

[30]  Paul M. Thompson,et al.  Intrinsic brain surface conformal mapping using a variational method , 2004, SPIE Medical Imaging.

[31]  K. Svoboda,et al.  An alternative semi-implicit Euler method for the integration of highly stiff nonlinear differential equations , 1997 .

[32]  David A. Rottenberg,et al.  Quasi-Conformally Flat Mapping the Human Cerebellum , 1999, MICCAI.